Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{-t^2 - 10t - 16}{-5t^2 + 10t + 400}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ q = \dfrac {-1(t^2 + 10t + 16)} {-5(t^2 - 2t - 80)} $ $ q = \dfrac{1}{5} \cdot \dfrac{t^2 + 10t + 16}{t^2 - 2t - 80} $ Next factor the numerator and denominator. $ q = \dfrac{1}{5} \cdot \dfrac{(t + 8)(t + 2)}{(t + 8)(t - 10)}$ Assuming $t \neq -8$ , we can cancel the $t + 8$ $ q = \dfrac{1}{5} \cdot \dfrac{t + 2}{t - 10}$ Therefore: $ q = \dfrac{ t + 2 }{ 5(t - 10)}$, $t \neq -8$